op amps have extremely high inherent gain (an "ideal" op amp has infinite gain!); you add components to limit gain to where you want it.It's the other parameters you have to look at: power supply, whether the inputs are allowed to go to the limits that your signal does, whether the outputs go the limits of the A-D input, and so on.

So a "real" opamp might have an open-loop gain of 250000 and easily configured to a gain of 2500, but if powered from 0/5V the input signal might need to be between 1.5 and 3.5 V, and the output may only travel between 1 and 4V...

Check out the wikipedia article. It's a start: http://en.wikipedia.org/wiki/Operational_amplifier

one would need to also need to center the voltage around 2.5V, and I believe there are chips to do that too.

Yes they are called op-amps.

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You need x2500 gain.

In practice this is very high and difficult to achieve due to the DC offset of the op amp. This is a small DC voltage that gets amplified at the same rate as the other signals and is dependent on the actual op-amp. It is normally in the mV range and so will swamp this signal.